## Kodai Mathematical Journal

### $L^2$ continuity of the Calderón type commutator for the Littlewood-Paley operator with rough variable kernel

#### Abstract

For $b \in Lip(\mathbf{R}^{n})$, the Calderón type commutator for the Littlewood-Paley operator with variable kernel is defined by $$\mu_{\Omega,1;b}(f)(x) = \left( \int_{0}^{\infty}\left| \frac{1}{t^2}\int_{\vert x-y\vert \leq t}\frac{\Omega(x, x-y)}{\vert x - y \vert ^{n-1}} (b(x) - b(y))f(y) dy\right|^{2} \frac{dt}{t}\right)^{1/2}.$$ By giving a method based on Littlewood-Paley theory, Fourier transform and the spherical harmonic development, we prove the $L^{2}$ norm inequalities for the rough operators $\mu_{\Omega,1;b}$ with $\Omega(x,z^{\prime}) \in L^{\infty}(\mathbf{R}^{n} \times L^{q}(S^{n-1})\left(q \gt \frac{2(n-1)}{n} \right)$ satisfying certain cancellation conditions.

#### Note

The research is supported by NSF of China (Grant: 11471033), NCET of China (Grant: NCET- 11-0574), the Fundamental Research Funds for the Central Universities (FRF-BR-16-011A).

#### Article information

Source
Kodai Math. J., Volume 40, Number 3 (2017), 405-420.

Dates
Revised: 24 November 2016
First available in Project Euclid: 31 October 2017

https://projecteuclid.org/euclid.kmj/1509415223

Digital Object Identifier
doi:10.2996/kmj/1509415223

Mathematical Reviews number (MathSciNet)
MR3718490

Zentralblatt MATH identifier
06827096

#### Citation

Chen, Yanping; Niu, Zhendong; Wang, Liwei. $L^2$ continuity of the Calderón type commutator for the Littlewood-Paley operator with rough variable kernel. Kodai Math. J. 40 (2017), no. 3, 405--420. doi:10.2996/kmj/1509415223. https://projecteuclid.org/euclid.kmj/1509415223